70 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV. 



with the position of P. This formula holds of course only so long 

 as the circle embraces the same internal boundary, and lies itself 

 wholly in the fluid. 



If the region be unlimited externally, and if the circle embrace 

 the whole of the internal boundaries, and if further the velocity be 

 everywhere zero at infinity, then C is an absolute constant ; as is 

 seen by reasoning similar to that of Art. 46. 



It may then be shewn, exactly as in Art. 51, that the value 

 of (j&amp;gt; at a very great distance r from the internal boundary tends 

 to the value If log r + C. In the particular case when M the 

 limit to which &amp;lt; tends at infinity is finite ; in all other cases it is 

 infinite, and of the same sign as M. 



We infer, as in Art. 52, that there is only one single-valued 

 function &amp;lt; which (a) satisfies the equation (7) at every point of 

 the plane xy external to a given system of closed curves, (b) makes 



the value of ~ equal to an arbitrarily given quantity at every 



point of these curves, and (c) has its first differential coefficients 

 all zero at infinity. 



72. The kinetic energy of a portion of fluid bounded by a 

 cylindrical surface whose generating lines are parallel to the 

 axis of &, and two planes perpendicular to the axis of z at unit 

 distance, is given by the formula 



where the surface-integral is taken over the portion of the 

 plane xy cut off by the cylindrical surface, and the line-integral 

 round the boundary of this portion. 



If the cylindrical part of the boundary consist of two or more 

 separate portions one of which embraces all the rest, the enclosed 

 region is multiply -connected, and the equation (10) needs a 

 correction, which may be applied exactly as in Art. 66. 



If we attempt, by a process similar to that of Art. 65, to 

 calculate the energy in the case where the region extends to 

 infinity, we find that its value is infinite, except when 7l/ = 0. 



